Summary: Based on the preimage structure of the system \((X, T)\), M. Hurley [Ergodic Theory Dyn. Syst. 15, No. 3, 557–568 (1995; Zbl 0833.54021)] introduced the notion of pointwise topological preimage entropies \(h_m(T)\) and \(h_p(T)\). Furthermore, from the measure-theoretic point of view, W. Wu and Y. Zhu [Adv. Math. 406, Article ID 108483, 45 p. (2022; Zbl 1502.37044)] introduced a notion of pointwise metric preimage entropy \(h_{m, \mu}(T)\) for a \(T\)-invariant measure \(\mu\) on \(X\), and obtained the variational principle between \(h_{m, \mu}(T)\) and \(h_m(T)\) under the condition of uniform separation of preimages. A natural question is whether a variational principle for \(h_m(T)\) and \(h_{m, \mu}(T)\) without any additional assumptions. In this paper, we define a new version of topological preimage entropy \(h_m(T| \mu)\) relative to a \(T\)-invariant measure \(\mu\), and show that the inequality \(h_{m, \mu}(T) \leqslant h_m(T| \mu) \leqslant h_p(T)\) holds for every \(T\)-invariant probability measure \(\mu\). As a consequence, we obtain that there is a topological dynamical system \((X,T)\) such that the following strict inequality holds: \[ \sup_{\mu \in \mathcal{M}(X,T)} h_{m, \mu}(T) < h_m(T), \] where \(\mathcal{M}(X,T)\) denote the set of all \(T\)-invariant probability measures.